Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count

Abstract

The nodal domains of eigenvectors of the discrete Schrodinger operator on\nsimple, finite and connected graphs are considered. Courant's well known nodal\ndomain theorem applies in the present case, and sets an upper bound to the\nnumber of nodal domains of eigenvectors: Arranging the spectrum as a non\ndecreasing sequence, and denoting by $\\nu_n$ the number of nodal domains of the\n$n$'th eigenvector, Courant's theorem guarantees that the nodal deficiency\n$n-\\nu_n$ is non negative. (The above applies for generic eigenvectors. Special\ncare should be exercised for eigenvectors with vanishing components.) The main\nresult of the present work is that the nodal deficiency for generic\neigenvectors equals to a Morse index of an energy functional whose value at its\nrelevant critical points coincides with the eigenvalue. The association of the\nnodal deficiency to the stability of an energy functional at its critical\npoints was recently discussed in the context of quantum graphs\n[arXiv:1103.1423] and Dirichlet Laplacian in bounded domains in $R^d$\n[arXiv:1107.3489]. The present work adapts this result to the discrete case.\nThe definition of the energy functional in the discrete case requires a special\nsetting, substantially different from the one used in\n[arXiv:1103.1423,arXiv:1107.3489] and it is presented here in detail.\n